A stochastic differential equation SIS epidemic model with two independent Brownian motions

Cai, Siyang and Cai, Yongmei and Mao, Xuerong (2019) A stochastic differential equation SIS epidemic model with two independent Brownian motions. Journal of Mathematical Analysis and Applications, 474 (2). pp. 1536-1550. ISSN 0022-247X

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    Abstract

    In this paper, we introduce two perturbations in the classical deterministic susceptible–infected–susceptible epidemic model. Greenhalgh and Gray [4] in 2011 use a perturbation on β in SIS model. Based on their previous work, we consider another perturbation on the parameter μ+ γ and formulate the original model as a stochastic differential equation (SDE) with two independent Brownian Motions for the number of infected population. We then prove that our Model has a unique and bounded global solution I ( t ) . Also we establish conditions for extinction and persistence of the infected population I ( t ) . Under the conditions of persistence, we show that there is a unique stationary distribution and derive its mean and variance. Computer simulations illustrate our results and provide evidence to back up our theory.